Difference between kinetic energy and momentum?

I was wondering; what exactly is the difference between kinetic energy and momentum? I was just thinking about this the other day, and I couldn't quite work it out.

I'm not talking in definition-wise terms; I know that momentum=mv and kinetic energy=1/2 mv^2, and can see that the derivative of kinetic energy with respect to v gives you momentum.

My question is, what exactly is the physical relationship between these two, for lack of a better term, attributes of an object in motion. Like, what does squaring the velocity actually do to differentiate the two? I'm actually not too sure what it is I'm looking for, all I know is I haven't figured it out satisfactorally yet :P. I guess I'm just looking for the conceptual difference between them.

I'll try to explain this to the best of my knowledge, which is limited and firearm related.

Kinetic energy is a magnitude. It only describes how hard an object in motion will another object and is expressed in foot pounds. Momentum is the application of kinetic energy and is expressed in joules, foot pounds per second, etc..... When comparing moving objects that are similar (IE, bullets of different weights and velocities), the projectile with more momentum would tend to move its target father regardless of kinetic energy.
Example.... A rimfire rifle is fired at a steel plate at 50 yards. If you hit it, it flips over. You have a choice of using a rifle that will propel a 40 grain bullet at 1249 f/p/s, or a 17 grain bullet at 2199 f/p/s at 50 yards (these are actual figures). If you only used the formula for KE, you would find the bullets would provide 139 ft-lbs & 183 ft-lbs respectively, so you would likely pick the bullet with the higher KE for the task at hand. Looking into the figures a little more, by figuring momentum.......the 40 grain bullet would provide 7.14 ft-lbs/second, and the 17 grain bullet would provide 5.34 ft-lbs/second (weight/7000*velocity). The heavier, slower bullet would be more likely to flip the plate because it has more momentum even though the lighter faster bullet has 44 more ft-lbs of kinetic energy.

Hope this helps....

As for why figures are squared, lets just say that the formulas are hard enough to remember as it is. Knowing why will just confuse me more....:surprised

To add to the previous post, Kinetic Energy is a scalar (no specified direction but has a magnitude) while momentum is a vector (has a definite direction and magnitude). Their applications are different due to this distinction. Think of kinetic energy as a scalar quantity that can increase or decrease by changing the momentum of an object. Kinetic energy is useful because you don't have to say how an object is moving (and in what directions), you can just say that it has so much energy that is stored as the sum of its movement. In situations where the movement is complicated or even random, it may not be possible to speak of the momentum of particle X because you can't narrow down the direction but you can still perform measurements to figure out how much energy it has.

When there are collisions, the effect depends on mommentum of the bodies and not kinetic energy. similarly when two bodies are kept in contact with each other, the flow of heat energy depends on temperature and not heat energy. similar-amount of air and pressure, water level and amount of water.